Posted by sied on Thursday, October 10, 2013 at 7:44pm.
The height is L sin(theta). The decrease in the gravitational potential energy is thus m g L sin(theta).
The magnitude of the component of the gravitational force orthogonal to the incline is m g cos(theta), therefore the normal force is equal to
m g cos(theta). The friction force is thus equal to mu_1 m g cos(theta), the work done by the friction force is thus:
- mu_1 m g cos(theta) L
the minus sign coming from the fact that the force has a direction oposite to the displacement.
The kinetic energy at the bottom of the incline is thus given by:
m g L [sin(theta) - mu_1 cos(theta)]
Then there is a big mistake in the problem, because it assumes that the block will start to slide on the rough horizontal surface with with an initial velocity that corresponds to the above kinetic energy, but this wrong. I'll explain that later.
If the kinetic energy at the start of the horizntal rough surface is E, then you obviously have that the distance d it will travel satisfies:
mu_2 m g d = E
So, d = E/(mu_2 m g).
Now, you can't take E equal to
m g L [sin(theta) - mu_1 cos(theta)]
because the block has to change direction which requires an extra normal force and therefore additional friction forces. To calculate this effect, let's assume that the change in direction happens on the surface with coefficient of friction mu_1 over a very short distance at the ground level.
Then what happens is that the component of the block in the vertical direction will have to vanish when the incline levels off and the block is about to enter the rough surface. The integral of the component of the normal force in the vertical direction over time during the change of direction is thus equal to p sin(theta) where p is the magnitude of the initial momentum.
The friction force is always orthogonal to the normal force, and equal to mu_1 times that normal force. Therefore the component of the friction force in the horizontal direction is - mu_1 times the component if the normal force in the vertical direction, the integral over time during the change in direction is thus equal to
-mu_1 p sin(theta) and this is then the change in the component of the momentum in the horizontal direction.
The initial kinetic energy was:
E1 =
m g L [sin(theta) - mu_1 cos(theta)]
This is also given in terms of the momentum as:
E1 = p^2/(2m)
therefore:
|p| = sqrt(2 m E1)
the initial horizontal component is thus:
|p| cos(theta) = sqrt(2 m E1) cos(theta)
The final horizontal component is thus:
sqrt(2 m E1)*
[cos(theta) - mu_1 sin(theta)]
and this is the total momentum because the vertical component will have vanished as the incline levels off.
The kinetic energy at the start of the rough surface is thus:
E = E1 [cos(theta) - mu_1 sin(theta)]
The distance the block will slide is thus given by:
d = E/(mu_2 m g) =
L/mu_2 [cos(theta) - mu_1 sin(theta)]*
[sin(theta) - mu_1 cos(theta)] =
[1/2(1+mu_1^2)sin(2 theta)-mu_1]L/mu_2
Correction of the last part:
The kinetic energy at the start of the rough surface is thus:
E = E1 [cos(theta) - mu_1 sin(theta)]^2
The distance the block will slide is thus given by:
d = E/(mu_2 m g) =
L/mu_2 [cos(theta) - mu_1 sin(theta)]^2* [sin(theta) - mu_1 cos(theta)] =
[1/2(1+mu_1^2)sin(2 theta)-mu_1]*
[cos(theta) - mu_1 sin(theta)] L/mu_2
u r a genious man..!